On the range of the derivative of a real-valued function with bounded support
T. Gaspari
Studia Mathematica, Tome 151 (2002), p. 81-99 / Harvested from The Polish Digital Mathematics Library
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We study the set f’(X) = f’(x): x ∈ X when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a Cp-smooth bump b:X → ℝ such that ||b(p)||∞ is finite, then any connected open subset of X* containing 0 is the range of the derivative of a Cp-smooth bump. We also study the finite-dimensional case which is quite different. Finally, we show that in infinite-dimensional separable smooth Banach spaces, every analytic subset of X* which satisfies a natural linkage condition is the range of the derivative of a C¹-smooth bump. We then find an analogue of this condition in the finite-dimensional case

Publié le : 2002-01-01
EUDML-ID : urn:eudml:doc:284401 
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T. Gaspari. On the range of the derivative of a real-valued function with bounded support. Studia Mathematica, Tome 151 (2002) pp. 81-99. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-1-6/